Optimal extensions for $p$-th power factorable operators

Abstract

Let $X(\mu )$ be a function space related to a measure space $(\Omega ,\Sigma ,\mu )$ with ${\chi }_{\Omega }\in X(\mu )$ and let $T:X(\mu )\to E$ be a Banach space valued operator. It is known that if $T$ is $p$-th power factorable then the largest function space to which $T$ can be extended preserving $p$-th power factorability is given by the space $L^p(m_T)$ of $p$-integrable functions with respect to $m_T$, where $m_T:\Sigma \to E$ is the vector measure associated to $T$ via $m_T(A)=T({\chi }_A)$. In this paper we extend this result by removing the restriction ${\chi }_{\Omega }\in X(\mu )$. In this general case, by considering $m_T$ defined on a certain $\delta$-ring, we show that the optimal domain for $T$ is the space $L^p(m_T)\cap L^1(m_T)$. We apply the obtained results to the particular case when $T$ is a map between sequence spaces defined by an infinite matrix.